Substitute Gomis struck five minutes from time, meeting Jefferson Montero's cross with a header that crossed the line before it was scooped out by keeper David Ospina, the goal confirmed almost immediately to the referee by the goalline technology system.
Frenchman Gomis also scored the winner when Swansea beat Arsenal 2-1 at home in November.
Arsenal, who have three games left to play, remain on 70 points, three points behind City who have two games left. Manchester United, in fourth, have 68 points with the top three securing automatic entry into the Champions League group stage.
United host Arsenal at Old Trafford on Sunday.
"We were a bit unlucky with the goal. There are a lot of positives. This kind of game you play 20 times and win 19 times," Arsenal manager Arsene Wenger told Sky Sports.
"In patches our domination was top quality. You have to learn from defeat and we take encouragement from a defeat like that. Swansea deserve credit for their defensive display but they probably had the fewest passes in our half of any team here this season."
Swansea goalkeeper Lukasz Fabianski frustrated his former club, the Poland international making several good saves in the second half to thwart the home side.
Arsenal, unchanged for fifth successive game, had rarely tested Fabianski in a tepid opening half.
It took almost an hour for Arsenal to muster their first shot on target with Olivier Giroud shooting straight at Fabianski.
Arsenal pressed forward with greater urgency and Fabianski denied Alexis Sanchez and then Theo Walcott with a fine double save.
Swansea, who are in eighth place, are only the third team to beat Arsenal home and away in a league season in the Premier League era.
"We worked all week on our plan. The players were magnificent and carried it out to the letter," said Swansea manager Garry Monk.
